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Take an icosahedron, say one made out of something soft, like chocolate, and slice off each of the corners:

You get a structure called (rather prosaically) a truncated icosahedron. Here’s one rendered in wood (by artist Ai Weiwei):

This structure is also known as a soccer ball:

Or as buckminsterfullerene:

Continuing the Buckminster Fuller theme from last time, today’s column is devoted to notable truncated icosahedron constructions — this polyhedron has been used in an incredible variety of ways for decades.

First, there is the 1949 Autonomous Living Unit by Buckminster Fuller himself:

The great circles intersecting the faces of this structure make the truncated dodecahedron a bit hard to see, but it’s there: one of the pentagons is front and center, just above the white prism that may serve as the unit’s front door.

Second is the optional hub cap of the Pontiac Trans Am, model years 1971-76:

Third, featured at this year’s New York City Maker Faire at the New York Hall of Science, was the rotating Happy Fun Ball by Rob Marshall:

And finally, perhaps the most dramatic truncated icosahedron (or pair of them) yet, is Leo Villareal’s Buckyball, currently on display in Madison Square Park in Manhattan through 2013 Feb 1.

The striking edge lighting changes through a myriad of patterns:

It’s auspicious that Buckyball, which links mathematics and culture, will be on display less than 200 meters from the National Museum of Mathematics when it opens soon as America’s primary cultural institution centered on mathematics. We hope to see you in both places!